locally nilpotent group
Definition
A locally nilpotent group is a group in which every finitely generated subgroup is nilpotent.
Examples
All nilpotent groups are locally nilpotent, because subgroups of nilpotent groups are nilpotent.
An example of a locally nilpotent group that is not nilpotent is , the generalized dihedral group formed from the quasicyclic -group (http://planetmath.org/PGroup4) .
The Fitting subgroup of any group is locally nilpotent.
All N-groups are locally nilpotent. More generally, all Gruenberg groups are locally nilpotent.
Properties
Any subgroup or quotient (http://planetmath.org/QuotientGroup) of a locally nilpotent group is locally nilpotent. Restricted direct products of locally nilpotent groups are locally nilpotent.
For each prime , the elements of -power order in a locally nilpotent group form a fully invariant subgroup (the maximal -subgroup (http://planetmath.org/PGroup4)). The elements of finite order in a locally nilpotent group also form a fully invariant subgroup (the torsion subgroup), which is the restricted direct product of the maximal -subgroups. (This generalizes the fact that a finite nilpotent group is the direct product of its Sylow subgroups.)
Every group has a unique maximal locally nilpotent normal subgroup. This subgroup is called the Hirsch-Plotkin radical, or locally nilpotent radical, and is often denoted . If is finite (or, more generally, satisfies the maximal condition), then the Hirsch-Plotkin radical is the same as the Fitting subgroup, and is nilpotent.
Title | locally nilpotent group |
Canonical name | LocallyNilpotentGroup |
Date of creation | 2013-03-22 15:40:42 |
Last modified on | 2013-03-22 15:40:42 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 7 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20F19 |
Related topic | LocallyCalP |
Related topic | NilpotentGroup |
Related topic | NormalizerCondition |
Defines | locally nilpotent |
Defines | Hirsch-Plotkin radical |
Defines | locally nilpotent radical |